Standard +0.3 This is a standard Further Maths question on complex roots with real coefficients. Students must recognize that 1-2i is also a root (conjugate pair), form the quadratic factor (x-(1+2i))(x-(1-2i)) = x²-2x+5, perform polynomial division to find the remaining quadratic, then solve it. While it requires multiple steps and careful algebra, it follows a well-practiced procedure with no novel insight needed, making it slightly easier than average for Further Maths.
1.
$$f ( x ) = x ^ { 4 } - x ^ { 3 } - 9 x ^ { 2 } + 29 x - 60$$
Given that \(x = 1 + 2 \mathrm { i }\) is a root of the equation \(\mathrm { f } ( x ) = 0\), use algebra to find the three other roots of the equation \(\mathrm { f } ( x ) = 0\)
1.
$$f ( x ) = x ^ { 4 } - x ^ { 3 } - 9 x ^ { 2 } + 29 x - 60$$
Given that $x = 1 + 2 \mathrm { i }$ is a root of the equation $\mathrm { f } ( x ) = 0$, use algebra to find the three other roots of the equation $\mathrm { f } ( x ) = 0$\\
\hfill \mbox{\textit{Edexcel F1 2015 Q1 [7]}}